The document provides an introduction to time series regression and forecasting. It discusses key concepts like autocorrelation in time series data and the use of autoregressive (AR) models for forecasting. An example AR(1) model is estimated to predict inflation using the lagged change in inflation as a predictor. This shows the lagged variable is statistically significant but explains little of the variation. The document also demonstrates forecasting out-of-sample inflation values using the estimated AR(1) model.
INTRODUCTION TO TIME SERIES REGRESSION AND FORCASTINGSPICEGODDESS
What Is Time Series Regression? Time series regression is a statistical method for predicting a future response based on the response history (known as autoregressive dynamics) and the transfer of dynamics from relevant predictors.
Innovations in technology has revolutionized financial services to an extent that large financial institutions like Goldman Sachs are claiming to be technology companies! It is no secret that technological innovations like Data science and AI are changing fundamentally how financial products are created, tested and delivered. While it is exciting to learn about technologies themselves, there is very little guidance available to companies and financial professionals should retool and gear themselves towards the upcoming revolution.
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Topics in Econometrics
INTRODUCTION TO TIME SERIES REGRESSION AND FORCASTINGSPICEGODDESS
What Is Time Series Regression? Time series regression is a statistical method for predicting a future response based on the response history (known as autoregressive dynamics) and the transfer of dynamics from relevant predictors.
Innovations in technology has revolutionized financial services to an extent that large financial institutions like Goldman Sachs are claiming to be technology companies! It is no secret that technological innovations like Data science and AI are changing fundamentally how financial products are created, tested and delivered. While it is exciting to learn about technologies themselves, there is very little guidance available to companies and financial professionals should retool and gear themselves towards the upcoming revolution.
In this master class, we will discuss key innovations in Data Science and AI and connect applications of these novel fields in forecasting and optimization. Through case studies and examples, we will demonstrate why now is the time you should invest to learn about the topics that will reshape the financial services industry of the future!
Topics in Econometrics
Getting things right: optimal tax policy with labor market dualityGilbert Mbara
We develop a dynamic general equilibrium model in which firms evade the employer contribution component of social security taxes by offering some workers non-formal contracts. When calibrated, the model yields estimates of dual labor market participation consistent with empirical evidence for the EU14 countries and the US. We investigate the optimal mix of the avoidable and unavoidable components of labor taxes and analyze the fiscal and macroeconomics effects of bringing the composition to the welfare optimum. We find that partial labor tax evasion makes tax revenues more elastic, but full tax compliance is not necessarily a welfare enhancing policy mix.
Estimating Financial Frictions under LearningGRAPE
The paper studies the implication of initial beliefs and associated confidence under adaptive learning. We first illustrate how prior beliefs determine learning dynamics and the evolution of endogenous variables in a small DSGE model with credit-constrained agents, in which rational expectations are replaced by constant-gain adaptive learning. We then examine how discretionary experimenting with new macroeconomic policies is affected by expectations that agents have in relation to these policies. More specifically, we show that a newly introduced macro-prudential policy that aims at making leverage counter-cyclical can lead to substantial increase in fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect information about the policy experiment.
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The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
Scalable inference for a full multivariate stochastic volatilitySYRTO Project
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February 19, 2016
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International capital flows - data vs. theory
1 Feldstein-Horioka puzzle
• corr (S, I ) > 0 in the data
2 Lucas puzzle
• K has not flown to poor countries, despite
K
Y
poor
<
K
Y
rich
3 Allocation Puzzle
• corr (ΔTFP, Δexternal debt) < 0
4 Quantity Puzzle (not as famous as the other three)
• Neo-classical 1-sector model over-predicts international
capital flows by a factor of 10
• Gourinchas and Jeanne (REStud, 2013); Rothert (EL, 2016)
Non-tradable Goods, Factor Markets Frictions, and International Capital FlowsGRAPE
International capital flows - data vs. theory
1 Feldstein-Horioka puzzle
• corr (S, I ) > 0 in the data
2 Lucas puzzle
• K has not flown to poor countries, despite
K
Y
poor
<
K
Y
rich
3 Allocation Puzzle
• corr (ΔTFP, Δexternal debt) < 0
4 Quantity Puzzle (not as famous as the other three)
• Neo-classical 1-sector model over-predicts international
capital flows by a factor of 10
• Gourinchas and Jeanne (REStud, 2013); Rothert (EL, 2016)
Striking a balance: optimal tax policy with labor market dualityGRAPE
A DSGE model that explains cross country variation in labour market duality using the share of social security tax in the total labour tax wedge. Deterministic simulations show that a 'tax policy' reform that increases firms' share of the social security tax burden has a welfare enhancing effect across most EU countries.
HLEG thematic workshop on Economic Insecurity, Walter Bossert, presenterStatsCommunications
HLEG thematic workshop on Economic Insecurity, 4 March 2016, New York, United States. More information at: http://oecd/hleg-workshop-on-economic-insecurity-2016
Given a dataset, in the first question I evaluate the transmission mechanism of the monetary policy by means of a VAR model using as information set the vector of three variables yt = (Δgdpt , inflt , ratet)’, where Δgdpt is the growth rate of the gdp, inflt is the inflation rate and ratet is the policy interest rate. Using the first three series reported in the dataset, I specify and estimate a VAR model and discuss the monetary transmission mechanism by calculating structural impulse response functions. Moreover, I test whether the policy interest rate does cause (in the Granger sense) the other two variables in the system. Finally, I choose one of the three variables and estimate an appropriate ARMA model, commenting on the differences with respect to the corresponding equation in the VAR model.
In the second question, I verify that the three variables are nonstationary and, using the Engle-Granger approach, whether the series are effectively cointegrated. Moreover, I verify if and which variable reacts to the disequilibria from the cointegrating relation.
Striking a balance: optimal tax policy with labor market dualityGRAPE
We show that the cross country variation in the level of social security taxes explains the variation in the level of underground or 'secondary' labor market activities. We then perform policy experiments where we vary the level of this taxes, shifting most of the burden from workers to firms and vice-versa. We find that it is optimal to make firms take on more of the "labor income tax" wedge - both in terms of welfare and fiscal revenue.
Getting things right: optimal tax policy with labor market dualityGilbert Mbara
We develop a dynamic general equilibrium model in which firms evade the employer contribution component of social security taxes by offering some workers non-formal contracts. When calibrated, the model yields estimates of dual labor market participation consistent with empirical evidence for the EU14 countries and the US. We investigate the optimal mix of the avoidable and unavoidable components of labor taxes and analyze the fiscal and macroeconomics effects of bringing the composition to the welfare optimum. We find that partial labor tax evasion makes tax revenues more elastic, but full tax compliance is not necessarily a welfare enhancing policy mix.
Estimating Financial Frictions under LearningGRAPE
The paper studies the implication of initial beliefs and associated confidence under adaptive learning. We first illustrate how prior beliefs determine learning dynamics and the evolution of endogenous variables in a small DSGE model with credit-constrained agents, in which rational expectations are replaced by constant-gain adaptive learning. We then examine how discretionary experimenting with new macroeconomic policies is affected by expectations that agents have in relation to these policies. More specifically, we show that a newly introduced macro-prudential policy that aims at making leverage counter-cyclical can lead to substantial increase in fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect information about the policy experiment.
The dangers of policy experiments Initial beliefs under adaptive learningGRAPE
The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
Scalable inference for a full multivariate stochastic volatilitySYRTO Project
Scalable inference for a full multivariate stochastic volatility
P. Dellaportas, A. Plataniotis and M. Titsias UCL(London), AUEB(Athens), AUEB(Athens)
Final SYRTO Conference - Université Paris1 Panthéon-Sorbonne
February 19, 2016
Non-tradable Goods, Factor Markets Frictions, and International Capital FlowsGRAPE
International capital flows - data vs. theory
1 Feldstein-Horioka puzzle
• corr (S, I ) > 0 in the data
2 Lucas puzzle
• K has not flown to poor countries, despite
K
Y
poor
<
K
Y
rich
3 Allocation Puzzle
• corr (ΔTFP, Δexternal debt) < 0
4 Quantity Puzzle (not as famous as the other three)
• Neo-classical 1-sector model over-predicts international
capital flows by a factor of 10
• Gourinchas and Jeanne (REStud, 2013); Rothert (EL, 2016)
Non-tradable Goods, Factor Markets Frictions, and International Capital FlowsGRAPE
International capital flows - data vs. theory
1 Feldstein-Horioka puzzle
• corr (S, I ) > 0 in the data
2 Lucas puzzle
• K has not flown to poor countries, despite
K
Y
poor
<
K
Y
rich
3 Allocation Puzzle
• corr (ΔTFP, Δexternal debt) < 0
4 Quantity Puzzle (not as famous as the other three)
• Neo-classical 1-sector model over-predicts international
capital flows by a factor of 10
• Gourinchas and Jeanne (REStud, 2013); Rothert (EL, 2016)
Striking a balance: optimal tax policy with labor market dualityGRAPE
A DSGE model that explains cross country variation in labour market duality using the share of social security tax in the total labour tax wedge. Deterministic simulations show that a 'tax policy' reform that increases firms' share of the social security tax burden has a welfare enhancing effect across most EU countries.
HLEG thematic workshop on Economic Insecurity, Walter Bossert, presenterStatsCommunications
HLEG thematic workshop on Economic Insecurity, 4 March 2016, New York, United States. More information at: http://oecd/hleg-workshop-on-economic-insecurity-2016
Given a dataset, in the first question I evaluate the transmission mechanism of the monetary policy by means of a VAR model using as information set the vector of three variables yt = (Δgdpt , inflt , ratet)’, where Δgdpt is the growth rate of the gdp, inflt is the inflation rate and ratet is the policy interest rate. Using the first three series reported in the dataset, I specify and estimate a VAR model and discuss the monetary transmission mechanism by calculating structural impulse response functions. Moreover, I test whether the policy interest rate does cause (in the Granger sense) the other two variables in the system. Finally, I choose one of the three variables and estimate an appropriate ARMA model, commenting on the differences with respect to the corresponding equation in the VAR model.
In the second question, I verify that the three variables are nonstationary and, using the Engle-Granger approach, whether the series are effectively cointegrated. Moreover, I verify if and which variable reacts to the disequilibria from the cointegrating relation.
Striking a balance: optimal tax policy with labor market dualityGRAPE
We show that the cross country variation in the level of social security taxes explains the variation in the level of underground or 'secondary' labor market activities. We then perform policy experiments where we vary the level of this taxes, shifting most of the burden from workers to firms and vice-versa. We find that it is optimal to make firms take on more of the "labor income tax" wedge - both in terms of welfare and fiscal revenue.
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Ch 12 Slides.doc. Introduction of science of business
1. 12-1
Introduction to Time Series Regression and
Forecasting
(SW Chapter 12)
Time series data are data collected on the same
observational unit at multiple time periods
Aggregate consumption and GDP for a country (for
example, 20 years of quarterly observations = 80
observations)
Yen/$, pound/$ and Euro/$ exchange rates (daily data
for 1 year = 365 observations)
Cigarette consumption per capital for a state
4. 12-4
Why use time series data?
To develop forecasting models
oWhat will the rate of inflation be next year?
To estimate dynamic causal effects
oIf the Fed increases the Federal Funds rate now,
what will be the effect on the rates of inflation and
unemployment in 3 months? in 12 months?
oWhat is the effect over time on cigarette
consumption of a hike in the cigarette tax
Plus, sometimes you don’t have any choice…
oRates of inflation and unemployment in the US can
be observed only over time.
5. 12-5
Time series data raises new technical issues
Time lags
Correlation over time (serial correlation or
autocorrelation)
Forecasting models that have no causal interpretation
(specialized tools for forecasting):
oautoregressive (AR) models
oautoregressive distributed lag (ADL) models
Conditions under which dynamic effects can be
estimated, and how to estimate them
Calculation of standard errors when the errors are
serially correlated
6. 12-6
Using Regression Models for Forecasting
(SW Section 12.1)
Forecasting and estimation of causal effects are quite
different objectives.
For forecasting,
o 2
R matters (a lot!)
oOmitted variable bias isn’t a problem!
oWe will not worry about interpreting coefficients
in forecasting models
oExternal validity is paramount: the model
estimated using historical data must hold into the
(near) future
7. 12-7
Introduction to Time Series Data
and Serial Correlation
(SW Section 12.2)
First we must introduce some notation and terminology.
Notation for time series data
Yt = value of Y in period t.
Data set: Y1,…,YT = T observations on the time series
random variable Y
We consider only consecutive, evenly-spaced
observations (for example, monthly, 1960 to 1999, no
missing months) (else yet more complications...)
8. 12-8
We will transform time series variables using lags,
first differences, logarithms, & growth rates
9. 12-9
Example: Quarterly rate of inflation at an annual rate
CPI in the first quarter of 1999 (1999:I) = 164.87
CPI in the second quarter of 1999 (1999:II) = 166.03
Percentage change in CPI, 1999:I to 1999:II
=
166.03 164.87
100
164.87
=
1.16
100
164.87
= 0.703%
Percentage change in CPI, 1999:I to 1999:II, at an
annual rate = 4 0.703 = 2.81% (percent per year)
Like interest rates, inflation rates are (as a matter of
convention) reported at an annual rate.
Using the logarithmic approximation to percent changes
yields 4 100 [log(166.03) – log(164.87)] = 2.80%
10. 12-10
Example: US CPI inflation – its first lag and its change
CPI = Consumer price index (Bureau of Labor Statistics)
11. 12-11
Autocorrelation
The correlation of a series with its own lagged values is
called autocorrelation or serial correlation.
The first autocorrelation of Yt is corr(Yt,Yt–1)
The first autocovariance of Yt is cov(Yt,Yt–1)
Thus
corr(Yt,Yt–1) = 1
1
cov( , )
var( )var( )
t t
t t
Y Y
Y Y
=1
These are population correlations – they describe the
population joint distribution of (Yt,Yt–1)
13. 12-13
Sample autocorrelations
The jth
sample autocorrelation is an estimate of the jth
population autocorrelation:
ˆ j
=
cov( , )
var( )
t t j
t
Y Y
Y
where
cov( , )
t t j
Y Y = 1, 1,
1
1
( )( )
1
T
t j T t j T j
t j
Y Y Y Y
T j
where 1,
j T
Y is the sample average of Yt computed over
observations t = j+1,…,T
oNote: the summation is over t=j+1 to T (why)?
15. 12-15
The inflation rate is highly serially correlated (1 = .85)
Last quarter’s inflation rate contains much information
about this quarter’s inflation rate
The plot is dominated by multiyear swings
But there are still surprise movements!
18. 12-18
Stationarity: a key idea for external validity of time
series regression
Stationarity says that the past is like the present and
the future, at least in a probabilistic sense.
We’ll focus on the case that Yt stationary.
19. 12-19
Autoregressions
(SW Section 12.3)
A natural starting point for a forecasting model is to use
past values of Y (that is, Yt–1, Yt–2,…) to forecast Yt.
An autoregression is a regression model in which Yt
is regressed against its own lagged values.
The number of lags used as regressors is called the
order of the autoregression.
oIn a first order autoregression, Yt is regressed
against Yt–1
oIn a pth
order autoregression, Yt is regressed
against Yt–1,Yt–2,…,Yt–p.
20. 12-20
The First Order Autoregressive (AR(1)) Model
The population AR(1) model is
Yt = 0 + 1Yt–1 + ut
0 and 1 do not have causal interpretations
if 1 = 0, Yt–1 is not useful for forecasting Yt
The AR(1) model can be estimated by OLS regression
of Yt against Yt–1
Testing 1 = 0 v. 1 0 provides a test of the
hypothesis that Yt–1 is not useful for forecasting Yt
21. 12-21
Example: AR(1) model of the change in inflation
Estimated using data from 1962:I – 1999:IV:
t
Inf
= 0.02 – 0.211Inft–1
2
R = 0.04
(0.14) (0.106)
Is the lagged change in inflation a useful predictor of the
current change in inflation?
t = .211/.106 = 1.99 > 1.96
Reject H0: 1 = 0 at the 5% significance level
Yes, the lagged change in inflation is a useful
predictor of current change in infl. (but low 2
R !)
22. 12-22
Example: AR(1) model of inflation – STATA
First, let STATA know you are using time series data
generate time=q(1959q1)+_n-1; _n is the observation no.
So this command creates a new variable
time that has a special quarterly
date format
format time %tq; Specify the quarterly date format
sort time; Sort by time
tsset time; Let STATA know that the variable time
is the variable you want to indicate the
time scale
23. 12-23
Example: AR(1) model of inflation – STATA, ctd.
. gen lcpi = log(cpi); variable cpi is already in memory
. gen inf = 400*(lcpi[_n]-lcpi[_n-1]); quarterly rate of inflation at an
annual rate
. corrgram inf , noplot lags(8); computes first 8 sample autocorrelations
LAG AC PAC Q Prob>Q
-----------------------------------------
1 0.8459 0.8466 116.64 0.0000
2 0.7663 0.1742 212.97 0.0000
3 0.7646 0.3188 309.48 0.0000
4 0.6705 -0.2218 384.18 0.0000
5 0.5914 0.0023 442.67 0.0000
6 0.5538 -0.0231 494.29 0.0000
7 0.4739 -0.0740 532.33 0.0000
8 0.3670 -0.1698 555.3 0.0000
. gen inf = 400*(lcpi[_n]-lcpi[_n-1])
This syntax creates a new variable, inf, the “nth” observation of which is
400 times the difference between the nth observation on lcpi and the “n-
1”th observation on lcpi, that is, the first difference of lcpi
24. 12-24
Example: AR(1) model of inflation – STATA, ctd
Syntax: L.dinf is the first lag of dinf
. reg dinf L.dinf if tin(1962q1,1999q4), r;
Regression with robust standard errors Number of obs = 152
F( 1, 150) = 3.96
Prob > F = 0.0484
R-squared = 0.0446
Root MSE = 1.6619
------------------------------------------------------------------------------
| Robust
dinf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dinf |
L1 | -.2109525 .1059828 -1.99 0.048 -.4203645 -.0015404
_cons | .0188171 .1350643 0.14 0.889 -.2480572 .2856914
------------------------------------------------------------------------------
if tin(1962q1,1999q4)
STATA time series syntax for using only observations between 1962q1 and
1999q4 (inclusive).
This requires defining the time scale first, as we did above
25. 12-25
Forecasts and forecast errors
A note on terminology:
A predicted value refers to the value of Y predicted
(using a regression) for an observation in the sample
used to estimate the regression – this is the usual
definition
A forecast refers to the value of Y forecasted for an
observation not in the sample used to estimate the
regression.
Predicted values are “in sample”
Forecasts are forecasts of the future – which cannot
have been used to estimate the regression.
26. 12-26
Forecasts: notation
Yt|t–1 = forecast of Yt based on Yt–1,Yt–2,…, using the
population (true unknown) coefficients
| 1
ˆ
t t
Y = forecast of Yt based on Yt–1,Yt–2,…, using the
estimated coefficients, which were estimated using
data through period t–1.
For an AR(1),
Yt|t–1 = 0 + 1Yt–1
| 1
ˆ
t t
Y = 0
ˆ
+ 1
ˆ
Yt–1, where 0
ˆ
and 1
ˆ
were estimated
using data through period t–1.
27. 12-27
Forecast errors
The one-period ahead forecast error is,
forecast error = Yt – | 1
ˆ
t t
Y
The distinction between a forecast error and a residual is
the same as between a forecast and a predicted value:
a residual is “in-sample”
a forecast error is “out-of-sample” – the value of Yt
isn’t used in the estimation of the regression
coefficients
28. 12-28
The root mean squared forecast error (RMSFE)
RMSFE = 2
| 1
ˆ
[( ) ]
t t t
E Y Y
The RMSFE is a measure of the spread of the forecast
error distribution.
The RMSFE is like the standard deviation of ut,
except that it explicitly focuses on the forecast error
using estimated coefficients, not using the population
regression line.
The RMSFE is a measure of the magnitude of a
typical forecasting “mistake”
29. 12-29
Example: forecasting inflation using and AR(1)
AR(1) estimated using data from 1962:I – 1999:IV:
t
Inf
= 0.02 – 0.211Inft–1
Inf1999:III = 2.8 (units are percent, at an annual rate)
Inf1999:IV = 3.2
Inf1999:IV = 0.4
So the forecast of Inf2000:I is,
2000: |1999:
I IV
Inf
= 0.02 – 0.211 0.4 = -0.06 -0.1
so
2000: |1999:
I IV
Inf = Inf1999:IV + 2000: |1999:
I IV
Inf
= 3.2 – 0.1 = 3.1
30. 12-30
The pth
order autoregressive model (AR(p))
Yt = 0 + 1Yt–1 + 2Yt–2 + … + pYt–p + ut
The AR(p) model uses p lags of Y as regressors
The AR(1) model is a special case
The coefficients do not have a causal interpretation
To test the hypothesis that Yt–2,…,Yt–p do not further
help forecast Yt, beyond Yt–1, use an F-test
Use t- or F-tests to determine the lag order p
Or, better, determine p using an “information criterion”
(see SW Section 12.5 – we won’t cover this)
31. 12-31
Example: AR(4) model of inflation
t
Inf
= .02 – .21Inft–1 – .32Inft–2 + .19Inft–3
(.12) (.10) (.09) (.09)
– .04Inft–4, 2
R = 0.21
(.10)
F-statistic testing lags 2, 3, 4 is 6.43 (p-value < .001)
2
R increased from .04 to .21 by adding lags 2, 3, 4
Lags 2, 3, 4 (jointly) help to predict the change in
inflation, above and beyond the first lag
32. 12-32
Example: AR(4) model of inflation – STATA
. reg dinf L(1/4).dinf if tin(1962q1,1999q4), r;
Regression with robust standard errors Number of obs = 152
F( 4, 147) = 6.79
Prob > F = 0.0000
R-squared = 0.2073
Root MSE = 1.5292
------------------------------------------------------------------------------
| Robust
dinf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dinf |
L1 | -.2078575 .09923 -2.09 0.038 -.4039592 -.0117558
L2 | -.3161319 .0869203 -3.64 0.000 -.4879068 -.144357
L3 | .1939669 .0847119 2.29 0.023 .0265565 .3613774
L4 | -.0356774 .0994384 -0.36 0.720 -.2321909 .1608361
_cons | .0237543 .1239214 0.19 0.848 -.2211434 .268652
------------------------------------------------------------------------------
NOTES
L(1/4).dinf is A convenient way to say “use lags 1–4 of dinf as regressors”
L1,…,L4 refer to the first, second,… 4th lags of dinf
33. 12-33
Example: AR(4) model of inflation – STATA, ctd.
. dis "Adjusted Rsquared = " _result(8); result(8) is the rbar-squared
Adjusted Rsquared = .18576822 of the most recently run regression
. test L2.dinf L3.dinf L4.dinf; L2.dinf is the second lag of dinf, etc.
( 1) L2.dinf = 0.0
( 2) L3.dinf = 0.0
( 3) L4.dinf = 0.0
F( 3, 147) = 6.43
Prob > F = 0.0004
Note: some of the time series features of STATA differ
between STATA v. 7 and STATA v. 8…
34. 12-34
Digression: we used Inf, not Inf, in the AR’s. Why?
The AR(1) model of Inft–1 is an AR(2) model of Inft:
Inft = 0 + 1Inft–1 + ut
or
Inft – Inft–1 = 0 + 1(Inft–1 – Inft–2) + ut
or
Inft = Inft–1 + 0 + 1Inft–1 – 1Inft–2 + ut
so
Inft = 0 + (1+1)Inft–1 – 1Inft–2 + ut
So why use Inft, not Inft?
35. 12-35
AR(1) model of Inf: Inft = 0 + 1Inft–1 + ut
AR(2) model of Inf: Inft = 0 + 1Inft + 2Inft–1 + vt
When Yt is strongly serially correlated, the OLS
estimator of the AR coefficient is biased towards zero.
In the extreme case that the AR coefficient = 1, Yt isn’t
stationary: the ut’s accumulate and Yt blows up.
If Yt isn’t stationary, our regression theory are working
with here breaks down
Here, Inft is strongly serially correlated – so to keep
ourselves in a framework we understand, the
regressions are specified using Inf
For optional reading, see SW Section 12.6, 14.3, 14.4
36. 12-36
Time Series Regression with Additional Predictors
and the Autoregressive Distributed Lag (ADL) Model
(SW Section 12.4)
So far we have considered forecasting models that use
only past values of Y
It makes sense to add other variables (X) that might be
useful predictors of Y, above and beyond the predictive
value of lagged values of Y:
Yt = 0 + 1Yt–1 + … + pYt–p
+ 1Xt–1 + … + rXt–r + ut
This is an autoregressive distributed lag (ADL) model
37. 12-37
Example: lagged unemployment and inflation
According to the “Phillips curve” says that if
unemployment is above its equilibrium, or “natural,”
rate, then the rate of inflation will increase.
That is, Inft should be related to lagged values of the
unemployment rate, with a negative coefficient
The rate of unemployment at which inflation neither
increases nor decreases is often called the “non-
accelerating rate of inflation” unemployment rate: the
NAIRU
Is this relation found in US economic data?
Can this relation be exploited for forecasting inflation?
39. 12-39
Example: ADL(4,4) model of inflation
t
Inf
= 1.32 – .36Inft–1 – .34Inft–2 + .07Inft–3 – .03Inft–4
(.47) (.09) (.10) (.08) (.09)
– 2.68Unemt–1 + 3.43Unemt–2 – 1.04Unemt–3 + .07Unempt–4
(.47) (.89) (.89) (.44)
2
R = 0.35 – a big improvement over the AR(4), for
which 2
R = .21
41. 12-41
Example: ADL(4,4) model of inflation – STATA, ctd.
. dis "Adjusted Rsquared = " _result(8);
Adjusted Rsquared = .34548812
. test L2.dinf L3.dinf L4.dinf;
( 1) L2.dinf = 0.0
( 2) L3.dinf = 0.0
( 3) L4.dinf = 0.0
F( 3, 143) = 4.93 The extra lags of dinf are signif.
Prob > F = 0.0028
. test L1.unem L2.unem L3.unem L4.unem;
( 1) L.unem = 0.0
( 2) L2.unem = 0.0
( 3) L3.unem = 0.0
( 4) L4.unem = 0.0
F( 4, 143) = 8.51 The lags of unem are significant
Prob > F = 0.0000
The null hypothesis that the coefficients on the lags of the unemployment
rate are all zero is rejected at the 1% significance level using the F-
statistic
42. 12-42
The test of the joint hypothesis that none of the X’s is a
useful predictor, above and beyond lagged values of Y, is
called a Granger causality test
“causality” is an unfortunate term here: Granger
Causality simply refers to (marginal) predictive content.
43. 12-43
Summary: Time Series Forecasting Models
For forecasting purposes, it isn’t important to have
coefficients with a causal interpretation!
Simple and reliable forecasts can be produced using
AR(p) models – these are common “benchmark”
forecasts against which more complicated forecasting
models can be assessed
Additional predictors (X’s) can be added; the result is
an autoregressive distributed lag (ADL) model
Stationary means that the models can be used outside
the range of data for which they were estimated
We now have the tools we need to estimate dynamic
causal effects...